Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1712 |
| commutator, Frattini, cosocle | C82 | C23 |
| prime radical | #1712 | C22 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(3,7)(4,10)(8,16)(9,14)(11,15)(12,13)(17,18,22,26)(19,27,29,41)(20,32,34,47)(21,35,33,25)(23,36,37,30)(24,39,40,28)(31,46,45,43)(38,44,42,48);(17,19,28,44,21,31,34,36,22,29,39,48,33,45,20,30)(18,23,32,43,25,38,40,41,26,37,47,46,35,42,24,27);(1,2,7,13,4,8,11,14,5,9,15,16,10,12,3,6)(19,29)(23,37)(27,41)(30,36)(31,45)(38,42)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=df,c^2=eg,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,c^b=ch86 conjugacy classes — view the full character table.
| maximal subgroup of | #72225191, #72225195, #72225196, … |
| maximal normal subgroup of | #72225191, #72225195, #72225196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30609);